| L(s) = 1 | + 0.107·2-s − 2.83·3-s − 1.98·4-s − 5-s − 0.304·6-s − 3.93·7-s − 0.428·8-s + 5.04·9-s − 0.107·10-s + 5.62·11-s + 5.63·12-s + 2.23·13-s − 0.421·14-s + 2.83·15-s + 3.93·16-s + 6.09·17-s + 0.540·18-s + 3.28·19-s + 1.98·20-s + 11.1·21-s + 0.603·22-s + 2.21·23-s + 1.21·24-s + 25-s + 0.240·26-s − 5.78·27-s + 7.81·28-s + ⋯ |
| L(s) = 1 | + 0.0758·2-s − 1.63·3-s − 0.994·4-s − 0.447·5-s − 0.124·6-s − 1.48·7-s − 0.151·8-s + 1.68·9-s − 0.0339·10-s + 1.69·11-s + 1.62·12-s + 0.620·13-s − 0.112·14-s + 0.732·15-s + 0.982·16-s + 1.47·17-s + 0.127·18-s + 0.754·19-s + 0.444·20-s + 2.43·21-s + 0.128·22-s + 0.462·23-s + 0.247·24-s + 0.200·25-s + 0.0470·26-s − 1.11·27-s + 1.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9257193927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9257193927\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 2 | \( 1 - 0.107T + 2T^{2} \) |
| 3 | \( 1 + 2.83T + 3T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 - 2.21T + 23T^{2} \) |
| 29 | \( 1 + 1.59T + 29T^{2} \) |
| 31 | \( 1 - 2.45T + 31T^{2} \) |
| 41 | \( 1 - 9.50T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 7.36T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 + 1.28T + 59T^{2} \) |
| 61 | \( 1 + 0.916T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 - 7.97T + 71T^{2} \) |
| 73 | \( 1 - 5.34T + 73T^{2} \) |
| 79 | \( 1 - 9.13T + 79T^{2} \) |
| 83 | \( 1 + 2.25T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 + 9.40T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77706756029368678989438455941, −7.08031715294560285429814665842, −6.31391758593724445769390137446, −5.89728514251047116669149192377, −5.31569429125690145454990150596, −4.28566296250637982891513633836, −3.83247532090614278035237912127, −3.14362352775706438784936602631, −1.01902852324213604204981004879, −0.74888009621036789531424634228,
0.74888009621036789531424634228, 1.01902852324213604204981004879, 3.14362352775706438784936602631, 3.83247532090614278035237912127, 4.28566296250637982891513633836, 5.31569429125690145454990150596, 5.89728514251047116669149192377, 6.31391758593724445769390137446, 7.08031715294560285429814665842, 7.77706756029368678989438455941
